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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a field is a set on which
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

addition
,
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

subtraction
,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
, and
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
are defined and behave as the corresponding operations on
rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογι ...
and
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s do. A field is thus a fundamental
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
which is widely used in
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
, and many other areas of mathematics. The best known fields are the field of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s, the field of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s and the field of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s. Many other fields, such as fields of rational functions,
algebraic function field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s,
algebraic number field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
. Most
cryptographic protocol A security protocol (cryptographic protocol or encryption protocol) is an abstract or concrete protocol that performs a security Security is freedom from, or resilience against, potential harm (or other unwanted Coercion, coercive change) caused b ...
s rely on
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a
field extension In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
.
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
, initiated by
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ...
in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that
angle trisection In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two rays lie in the plane (ge ...

angle trisection
and
squaring the circle Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "meas ...

squaring the circle
cannot be done with a
compass and straightedge Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandr ...
. Moreover, it shows that
quintic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
s are, in general, algebraically unsolvable. Fields serve as foundational notions in several mathematical domains. This includes different branches of
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, which is the standard general context for
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
.
Number field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, the siblings of the field of rational numbers, are studied in depth in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
. Function fields can help describe properties of geometric objects.


Definition

Informally, a field is a set, along with two operations defined on that set: an addition operation written as , and a multiplication operation written as , both of which behave similarly as they behave for
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s and
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, including the existence of an
additive inverse In mathematics, the additive inverse of a is the number that, when to , yields . This number is also known as the opposite (number), sign change, and negation. For a , it reverses its : the additive inverse (opposite number) of a is negative, ...
for all elements , and of a
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

multiplicative inverse
for every nonzero element . This allows one to also consider the so-called ''inverse'' operations of
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

subtraction
, , and
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
, , by defining: :, :.


Classic definition

Formally, a field is a set together with two
binary operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s on called ''addition'' and ''multiplication''. A binary operation on is a mapping , that is, a correspondence that associates with each ordered pair of elements of ''F'' a uniquely determined element of . The result of the addition of and is called the sum of and , and is denoted . Similarly, the result of the multiplication of and is called the product of and , and is denoted or . These operations are required to satisfy the following properties, referred to as ''
field axioms In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers do. A ...
'' (in these axioms, , , and are arbitrary
element Element may refer to: Science * Chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all ...
s of the field ): *
Associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...
of addition and multiplication: , and . *
Commutativity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

Commutativity
of addition and multiplication: , and . *
Additive Additive may refer to: Mathematics * Additive function In number theory, an additive function is an arithmetic function ''f''(''n'') of the positive integer ''n'' such that whenever ''a'' and ''b'' are coprime, the function of the product is the ...
and
multiplicative identity In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...
: there exist two different elements and in such that and . *
Additive inverse In mathematics, the additive inverse of a is the number that, when to , yields . This number is also known as the opposite (number), sign change, and negation. For a , it reverses its : the additive inverse (opposite number) of a is negative, ...
s: for every in , there exists an element in , denoted , called the ''additive inverse'' of , such that . *
Multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

Multiplicative inverse
s: for every in , there exists an element in , denoted by or , called the ''multiplicative inverse'' of , such that . *
Distributivity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of multiplication over addition: . This may be summarized by saying: a field has two operations, called addition and multiplication; it is an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
under addition with 0 as the additive identity; the nonzero elements are an abelian group under multiplication with 1 as the multiplicative identity; and multiplication distributes over addition. Even more summarized: a field is a
commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
where 0 \ne 1 and all nonzero elements are invertible.


Alternative definition

Fields can also be defined in different, but equivalent ways. One can alternatively define a field by four binary operations (addition, subtraction, multiplication, and division) and their required properties.
Division by zero In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
is, by definition, excluded. In order to avoid
existential quantifier In predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order l ...
s, fields can be defined by two binary operations (addition and multiplication), two unary operations (yielding the additive and multiplicative inverses respectively), and two
nullary Arity () is the number of argument of a function, arguments or operands taken by a function (mathematics), function or operation (mathematics), operation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', ...
operations (the constants and ). These operations are then subject to the conditions above. Avoiding existential quantifiers is important in
constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists. In classical mathematics, one can prove the existence of a mathematical object without "finding ...
and
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and software. It has sci ...

computing
. One may equivalently define a field by the same two binary operations, one unary operation (the multiplicative inverse), and two constants and , since and .The a priori twofold use of the symbol "−" for denoting one part of a constant and for the additive inverses is justified by this latter condition.


Examples


Rational numbers

Rational numbers have been widely used a long time before the elaboration of the concept of field. They are numbers that can be written as
fractions A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, ...
, where and are
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s, and . The additive inverse of such a fraction is , and the multiplicative inverse (provided that ) is , which can be seen as follows: : \frac b a \cdot \frac a b = \frac = 1. The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows: : \begin & \frac a b \cdot \left(\frac c d + \frac e f \right) \\
pt
pt
= & \frac a b \cdot \left(\frac c d \cdot \frac f f + \frac e f \cdot \frac d d \right) \\
pt
pt
= & \frac \cdot \left(\frac + \frac\right) = \frac \cdot \frac \\
pt
pt
= & \frac = \frac + \frac = \frac + \frac \\
pt
pt
= & \frac a b \cdot \frac c d + \frac a b \cdot \frac e f. \end


Real and complex numbers

The
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s , with the usual operations of addition and multiplication, also form a field. The
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s consist of expressions : with real, where is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad area ...
, i.e., a (non-real) number satisfying . Addition and multiplication of real numbers are defined in such a way that expressions of this type satisfy all field axioms and thus hold for . For example, the distributive law enforces : It is immediate that this is again an expression of the above type, and so the complex numbers form a field. Complex numbers can be geometrically represented as points in the
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
, with
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

Cartesian coordinates
given by the real numbers of their describing expression, or as the arrows from the origin to these points, specified by their length and an angle enclosed with some distinct direction. Addition then corresponds to combining the arrows to the intuitive parallelogram (adding the Cartesian coordinates), and the multiplication is – less intuitively – combining rotating and scaling of the arrows (adding the angles and multiplying the lengths). The fields of real and complex numbers are used throughout mathematics, physics, engineering, statistics, and many other scientific disciplines.


Constructible numbers

In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with
compass and straightedge Straightedge and compass construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandr ...
. For example, it was unknown to the Greeks that it is, in general, impossible to trisect a given angle in this way. These problems can be settled using the field of constructible numbers. Real constructible numbers are, by definition, lengths of line segments that can be constructed from the points 0 and 1 in finitely many steps using only
compass A compass is a device that shows the cardinal direction The four cardinal directions, or cardinal points, are the directions north, east, south, and west, commonly denoted by their initials N, E, S, and W. East and west are perpendicular ( ...

compass
and
straightedge A straightedge or straight edge is a tool used for drawing straight lines, or checking their straightness. If it has equally spaced markings along its length, it is usually called a ruler. Straightedges are used in the automotive service and mac ...

straightedge
. These numbers, endowed with the field operations of real numbers, restricted to the constructible numbers, form a field, which properly includes the field of rational numbers. The illustration shows the construction of
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

square root
s of constructible numbers, not necessarily contained within . Using the labeling in the illustration, construct the segments , , and a
semicircle In (and more specifically ), a semicircle is a one-dimensional of points that forms half of a . The full of a semicircle always measures 180° (equivalently, , or a ). It has only one line of symmetry (). In non-technical usage, the term "semi ...

semicircle
over (center at the
midpoint 282px, The midpoint of the segment (1, 1) to (2, 2) In geometry, the midpoint is the middle point (geometry), point of a line segment. It is Distance, equidistant from both endpoints, and it is the centroid both of the segment and of the endp ...

midpoint
), which intersects the
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

perpendicular
line through in a point , at a distance of exactly h=\sqrt p from when has length one. Not all real numbers are constructible. It can be shown that \sqrt 2 is not a constructible number, which implies that it is impossible to construct with compass and straightedge the length of the side of a
cube with volume 2
cube with volume 2
, another problem posed by the ancient Greeks.


A field with four elements

In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. The following example is a field consisting of four elements called , , , and . The notation is chosen such that plays the role of the additive identity element (denoted 0 in the axioms above), and is the multiplicative identity (denoted 1 in the axioms above). The field axioms can be verified by using some more field theory, or by direct computation. For example, : , which equals , as required by the distributivity. This field is called a
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
with four elements, and is denoted or . The subset consisting of and (highlighted in red in the tables at the right) is also a field, known as the ''
binary field (also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...
'' or . In the context of
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, a ...
and
Boolean algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, and are often denoted respectively by ''false'' and ''true'', the addition is then denoted
XOR Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false). It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, ...

XOR
(exclusive or), and the multiplication is denoted
AND And or AND may refer to: Logic, grammar, and computing * Conjunction (grammar) In grammar In linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study o ...
. In other words, the structure of the binary field is the basic structure that allows computing with
bit The bit is a basic unit of information in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algo ...
s.


Elementary notions

In this section, denotes an arbitrary field and and are arbitrary elements of .


Consequences of the definition

One has and . In particular, one may deduce the additive inverse of every element as soon as one knows . If then or must be 0, since, if , then . This means that every field is an
integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
. In addition, the following properties are true for any elements and : : : : : : if


The additive and the multiplicative group of a field

The axioms of a field imply that it is an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
under addition. This group is called the
additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structure ...
of the field, and is sometimes denoted by when denoting it simply as could be confusing. Similarly, the ''nonzero'' elements of form an abelian group under multiplication, called the
multiplicative group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and denoted by or just or . A field may thus be defined as set equipped with two operations denoted as an addition and a multiplication such that is an abelian group under addition, is an abelian group under multiplication (where 0 is the identity element of the addition), and multiplication is distributive over addition.Equivalently, a field is an
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of type , such that is not defined, and are abelian groups, and ⋅ is distributive over +.
Some elementary statements about fields can therefore be obtained by applying general facts of
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
. For example, the additive and multiplicative inverses and are uniquely determined by . The requirement follows, because 1 is the identity element of a group that does not contain 0. Thus, the
trivial ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique Ring (mathematics), ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any Rng (algebra)#Rng of square zer ...
, consisting of a single element, is not a field. Every finite subgroup of the multiplicative group of a field is
cyclic Cycle or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social scienc ...

cyclic
(see ).


Characteristic

In addition to the multiplication of two elements of ''F'', it is possible to define the product of an arbitrary element of by a positive
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
to be the -fold sum : (which is an element of .) If there is no positive integer such that :, then is said to have
characteristic Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
0. For example, the field of rational numbers has characteristic 0 since no positive integer is zero. Otherwise, if there ''is'' a positive integer satisfying this equation, the smallest such positive integer can be shown to be a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
. It is usually denoted by and the field is said to have characteristic then. For example, the field has characteristic 2 since (in the notation of the above addition table) . If has characteristic , then for all in . This implies that :, since all other
binomial coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s appearing in the
binomial formula In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of exponentiation, powers of a binomial (polynomial), binomial. According to the theorem, it is possible to expand the polynomial into a summati ...
are divisible by . Here, ( factors) is the -th power, i.e., the -fold product of the element . Therefore, the
Frobenius map In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (numbe ...
: is compatible with the addition in (and also with the multiplication), and is therefore a field homomorphism. The existence of this homomorphism makes fields in characteristic quite different from fields of characteristic 0.


Subfields and prime fields

A '' subfield'' of a field is a subset of that is a field with respect to the field operations of . Equivalently is a subset of that contains , and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. This means that , that for all both and are in , and that for all in , both and are in . Field homomorphisms are maps between two fields such that , , and , where and are arbitrary elements of . All field homomorphisms are
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. If is also
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, it is called an isomorphism (or the fields and are called isomorphic). A field is called a
prime field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
if it has no proper (i.e., strictly smaller) subfields. Any field contains a prime field. If the characteristic of is (a prime number), the prime field is isomorphic to the finite field introduced below. Otherwise the prime field is isomorphic to .


Finite fields

''Finite fields'' (also called ''Galois fields'') are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example is a field with four elements. Its subfield is the smallest field, because by definition a field has at least two distinct elements . The simplest finite fields, with prime order, are most directly accessible using
modular arithmetic #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ( ...
. For a fixed positive integer , arithmetic "modulo " means to work with the numbers : The addition and multiplication on this set are done by performing the operation in question in the set of integers, dividing by and taking the remainder as result. This construction yields a field precisely if is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
. For example, taking the prime results in the above-mentioned field . For and more generally, for any
composite number A composite number is a positive integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calcul ...
(i.e., any number which can be expressed as a product of two strictly smaller natural numbers), is not a field: the product of two non-zero elements is zero since in , which, as was explained above, prevents from being a field. The field with elements ( being prime) constructed in this way is usually denoted by . Every finite field has elements, where is prime and . This statement holds since may be viewed as a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
over its prime field. The
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
of this vector space is necessarily finite, say , which implies the asserted statement. A field with elements can be constructed as the
splitting field In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
of the polynomial :. Such a splitting field is an extension of in which the polynomial has zeros. This means has as many zeros as possible since the
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
of is . For , it can be checked case by case using the above multiplication table that all four elements of satisfy the equation , so they are zeros of . By contrast, in , has only two zeros (namely 0 and 1), so does not split into linear factors in this smaller field. Elaborating further on basic field-theoretic notions, it can be shown that two finite fields with the same order are isomorphic. It is thus customary to speak of ''the'' finite field with elements, denoted by or .


History

Historically, three algebraic disciplines led to the concept of a field: the question of solving polynomial equations,
algebraic number theory Algebraic number theory is a branch of number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is th ...
, and
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

algebraic geometry
. A first step towards the notion of a field was made in 1770 by
Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiacubic polynomial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
in the expression : (with being a third
root of unity The 5th roots of unity (blue points) in the complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are containe ...
) only yields two values. This way, Lagrange conceptually explained the classical solution method of
Scipione del Ferro Scipione del Ferro (6 February 1465 – 5 November 1526) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...
and
François Viète François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Gre ...
, which proceeds by reducing a cubic equation for an unknown to a quadratic equation for . Together with a similar observation for equations of degree 4, Lagrange thus linked what eventually became the concept of fields and the concept of groups. Vandermonde, also in 1770, and to a fuller extent,
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

Carl Friedrich Gauss
, in his ''
Disquisitiones Arithmeticae Title page of the first edition The (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through ...
'' (1801), studied the equation : for a prime and, again using modern language, the resulting cyclic
Galois group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. Gauss deduced that a regular -gon can be constructed if . Building on Lagrange's work,
Paolo Ruffini Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qua ...

Paolo Ruffini
claimed (1799) that
quintic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
s (polynomial equations of degree 5) cannot be solved algebraically; however, his arguments were flawed. These gaps were filled by
Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

Niels Henrik Abel
in 1824.
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ...
, in 1832, devised necessary and sufficient criteria for a polynomial equation to be algebraically solvable, thus establishing in effect what is known as
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
today. Both Abel and Galois worked with what is today called an
algebraic number field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, but conceived neither an explicit notion of a field, nor of a group. In 1871
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory In algebra, ring theory is the study of ring (mathematics), rings ...
introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the
German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of Germany, see also German nationality law * German language The German la ...
word ''Körper'', which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by . In 1881
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

Leopold Kronecker
defined what he called a ''domain of rationality'', which is a field of
rational fraction In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
s in modern terms. Kronecker's notion did not cover the field of all algebraic numbers (which is a field in Dedekind's sense), but on the other hand was more abstract than Dedekind's in that it made no specific assumption on the nature of the elements of a field. Kronecker interpreted a field such as abstractly as the rational function field . Prior to this, examples of transcendental numbers were known since
Joseph Liouville Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse L ...

Joseph Liouville
's work in 1844, until
Charles Hermite Charles Hermite () FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family Reso ...
(1873) and
Ferdinand von Lindemann Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
(1882) proved the transcendence of and , respectively. The first clear definition of an abstract field is due to . In particular,
Heinrich Martin Weber Heinrich Martin Weber (5 March 1842, Heidelberg, Germany ) , image_map = , map_caption = , map_width = 250px , capital = Berlin , coordinates = , largest_city = capital , languages_type = Official language , languages = German ...

Heinrich Martin Weber
's notion included the field F''p''.
Giuseppe Veronese Giuseppe Veronese (7 May 1854 – 17 July 1917) was an Italy, Italian mathematician. He was born in Chioggia, near Venice. Education Veronese earned his laurea in mathematics from the Istituto Tecnico di Venezia in 1872. Work Although Veronese's w ...

Giuseppe Veronese
(1891) studied the field of formal power series, which led to introduce the field of ''p''-adic numbers. synthesized the knowledge of abstract field theory accumulated so far. He axiomatically studied the properties of fields and defined many important field-theoretic concepts. The majority of the theorems mentioned in the sections
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
, #Constructing fields, Constructing fields and #Elementary notions, Elementary notions can be found in Steinitz's work. linked the notion of ordered field, orderings in a field, and thus the area of analysis, to purely algebraic properties. Emil Artin redeveloped Galois theory from 1928 through 1942, eliminating the dependency on the primitive element theorem.


Constructing fields


Constructing fields from rings

A
commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
is a set, equipped with an addition and multiplication operation, satisfying all the axioms of a field, except for the existence of multiplicative inverses . For example, the integers form a commutative ring, but not a field: the Multiplicative inverse, reciprocal of an integer is not itself an integer, unless . In the hierarchy of algebraic structures fields can be characterized as the commutative rings in which every nonzero element is a unit (ring theory), unit (which means every element is invertible). Similarly, fields are the commutative rings with precisely two distinct Ideal (ring theory), ideals, and . Fields are also precisely the commutative rings in which is the only prime ideal. Given a commutative ring , there are two ways to construct a field related to , i.e., two ways of modifying such that all nonzero elements become invertible: forming the field of fractions, and forming residue fields. The field of fractions of is , the rationals, while the residue fields of are the finite fields .


Field of fractions

Given an
integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, its field of fractions is built with the fractions of two elements of exactly as Q is constructed from the integers. More precisely, the elements of are the fractions where and are in , and . Two fractions and are equal if and only if . The operation on the fractions work exactly as for rational numbers. For example, :\frac+\frac = \frac. It is straightforward to show that, if the ring is an integral domain, the set of the fractions form a field. The field of the
rational fraction In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
s over a field (or an integral domain) is the field of fractions of the polynomial ring . The field of Laurent series :\sum_^\infty a_i x^i \ (k \in \Z, a_i \in F) over a field is the field of fractions of the ring of formal power series (in which ). Since any Laurent series is a fraction of a power series divided by a power of (as opposed to an arbitrary power series), the representation of fractions is less important in this situation, though.


Residue fields

In addition to the field of fractions, which embeds injective map, injectively into a field, a field can be obtained from a commutative ring by means of a surjective map onto a field . Any field obtained in this way is a quotient ring, quotient , where is a maximal ideal of . If local ring, has only one maximal ideal , this field is called the residue field of . The principal ideal, ideal generated by a single polynomial in the polynomial ring (over a field ) is maximal if and only if is irreducible polynomial, irreducible in , i.e., if cannot be expressed as the product of two polynomials in of smaller
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
. This yields a field : This field contains an element (namely the residue class of ) which satisfies the equation :. For example, is obtained from by adjunction (field theory), adjoining the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad area ...
symbol , which satisfies , where . Moreover, is irreducible over , which implies that the map that sends a polynomial to yields an isomorphism :\mathbf R[X]/\left(X^2 + 1\right) \ \stackrel \cong \longrightarrow \ \mathbf C.


Constructing fields within a bigger field

Fields can be constructed inside a given bigger container field. Suppose given a field , and a field containing as a subfield. For any element of , there is a smallest subfield of containing and , called the subfield of ''F'' generated by and denoted . The passage from to is referred to by ''adjunction (field theory), adjoining an element'' to . More generally, for a subset , there is a minimal subfield of containing and , denoted by . The compositum of two subfields and of some field is the smallest subfield of containing both and The compositum can be used to construct the biggest subfield of satisfying a certain property, for example the biggest subfield of , which is, in the language introduced below, algebraic over .Further examples include the maximal unramified extension or the maximal abelian extension within .


Field extensions

The notion of a subfield can also be regarded from the opposite point of view, by referring to being a ''
field extension In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
'' (or just extension) of , denoted by :, and read " over ". A basic datum of a field extension is its degree of a field extension, degree , i.e., the dimension of as an -vector space. It satisfies the formula :. Extensions whose degree is finite are referred to as finite extensions. The extensions and are of degree 2, whereas is an infinite extension.


Algebraic extensions

A pivotal notion in the study of field extensions are algebraic elements. An element is ''algebraic'' over if it is a zero of a function, root of a polynomial with coefficients in , that is, if it satisfies a polynomial equation :, with in , and . For example, the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad area ...
in is algebraic over , and even over , since it satisfies the equation :. A field extension in which every element of is algebraic over is called an algebraic extension. Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula. The subfield generated by an element , as above, is an algebraic extension of if and only if is an algebraic element. That is to say, if is algebraic, all other elements of are necessarily algebraic as well. Moreover, the degree of the extension , i.e., the dimension of as an -vector space, equals the minimal degree such that there is a polynomial equation involving , as above. If this degree is , then the elements of have the form :\sum_^ a_k x^k, \ \ a_k \in E. For example, the field of Gaussian rationals is the subfield of consisting of all numbers of the form where both and are rational numbers: summands of the form (and similarly for higher exponents) don't have to be considered here, since can be simplified to .


Transcendence bases

The above-mentioned field of
rational fraction In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
s , where is an indeterminate (variable), indeterminate, is not an algebraic extension of since there is no polynomial equation with coefficients in whose zero is . Elements, such as , which are not algebraic are called Algebraic element, transcendental. Informally speaking, the indeterminate and its powers do not interact with elements of . A similar construction can be carried out with a set of indeterminates, instead of just one. Once again, the field extension discussed above is a key example: if is not algebraic (i.e., is not a root of a function, root of a polynomial with coefficients in ), then is isomorphic to . This isomorphism is obtained by substituting to in rational fractions. A subset of a field is a transcendence basis if it is algebraically independent (don't satisfy any polynomial relations) over and if is an algebraic extension of . Any field extension has a transcendence basis. Thus, field extensions can be split into ones of the form (transcendental extension, purely transcendental extensions) and algebraic extensions.


Closure operations

A field is algebraically closed if it does not have any strictly bigger algebraic extensions or, equivalently, if any polynomial equation :, with coefficients , has a solution . By the fundamental theorem of algebra, is algebraically closed, i.e., ''any'' polynomial equation with complex coefficients has a complex solution. The rational and the real numbers are ''not'' algebraically closed since the equation : does not have any rational or real solution. A field containing is called an ''algebraic closure'' of if it is algebraic extension, algebraic over (roughly speaking, not too big compared to ) and is algebraically closed (big enough to contain solutions of all polynomial equations). By the above, is an algebraic closure of . The situation that the algebraic closure is a finite extension of the field is quite special: by the Artin-Schreier theorem, the degree of this extension is necessarily 2, and is elementarily equivalent to . Such fields are also known as real closed fields. Any field has an algebraic closure, which is moreover unique up to (non-unique) isomorphism. It is commonly referred to as ''the'' algebraic closure and denoted . For example, the algebraic closure of is called the field of algebraic numbers. The field is usually rather implicit since its construction requires the ultrafilter lemma, a set-theoretic axiom that is weaker than the axiom of choice. In this regard, the algebraic closure of , is exceptionally simple. It is the union of the finite fields containing (the ones of order ). For any algebraically closed field of characteristic 0, the algebraic closure of the field of Laurent series is the field of Puiseux series, obtained by adjoining roots of .


Fields with additional structure

Since fields are ubiquitous in mathematics and beyond, several refinements of the concept have been adapted to the needs of particular mathematical areas.


Ordered fields

A field ''F'' is called an ''ordered field'' if any two elements can be compared, so that and whenever and . For example, the real numbers form an ordered field, with the usual ordering . The Artin-Schreier theorem states that a field can be ordered if and only if it is a formally real field, which means that any quadratic equation :x_1^2 + x_2^2 + \dots + x_n^2 = 0 only has the solution . The set of all possible orders on a fixed field is isomorphic to the set of ring homomorphisms from the Witt ring (forms), Witt ring of quadratic forms over , to . An Archimedean field is an ordered field such that for each element there exists a finite expression : whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no infinitesimals (elements smaller than all rational numbers); or, yet equivalent, the field is isomorphic to a subfield of . An ordered field is Dedekind-complete if all upper bounds, lower bounds (see Dedekind cut) and limits, which should exist, do exist. More formally, each bounded set, bounded subset of is required to have a least upper bound. Any complete field is necessarily Archimedean, since in any non-Archimedean field there is neither a greatest infinitesimal nor a least positive rational, whence the sequence , every element of which is greater than every infinitesimal, has no limit. Since every proper subfield of the reals also contains such gaps, is the unique complete ordered field, up to isomorphism. Several foundational results in calculus follow directly from this characterization of the reals. The hyperreals form an ordered field that is not Archimedean. It is an extension of the reals obtained by including infinite and infinitesimal numbers. These are larger, respectively smaller than any real number. The hyperreals form the foundational basis of non-standard analysis.


Topological fields

Another refinement of the notion of a field is a topological field, in which the set is a topological space, such that all operations of the field (addition, multiplication, the maps and ) are continuous maps with respect to the topology of the space. The topology of all the fields discussed below is induced from a metric (mathematics), metric, i.e., a function : that measures a ''distance'' between any two elements of . The completion (metric space), completion of is another field in which, informally speaking, the "gaps" in the original field are filled, if there are any. For example, any irrational number , such as , is a "gap" in the rationals in the sense that it is a real number that can be approximated arbitrarily closely by rational numbers , in the sense that distance of and given by the absolute value is as small as desired. The following table lists some examples of this construction. The fourth column shows an example of a zero sequence, i.e., a sequence whose limit (for ) is zero. The field is used in number theory and p-adic analysis, -adic analysis. The algebraic closure carries a unique norm extending the one on , but is not complete. The completion of this algebraic closure, however, is algebraically closed. Because of its rough analogy to the complex numbers, it is sometimes called the field of Metric completions and algebraic closures, complex p-adic numbers and is denoted by .


Local fields

The following topological fields are called ''local fields'':Some authors also consider the fields and to be local fields. On the other hand, these two fields, also called Archimedean local fields, share little similarity with the local fields considered here, to a point that calls them "completely anomalous". * finite extensions of (local fields of characteristic zero) * finite extensions of , the field of Laurent series over (local fields of characteristic ). These two types of local fields share some fundamental similarities. In this relation, the elements and (referred to as uniformizer) correspond to each other. The first manifestation of this is at an elementary level: the elements of both fields can be expressed as power series in the uniformizer, with coefficients in . (However, since the addition in is done using carry (arithmetic), carrying, which is not the case in , these fields are not isomorphic.) The following facts show that this superficial similarity goes much deeper: * Any first order logic, first order statement that is true for almost all is also true for almost all . An application of this is the Ax-Kochen theorem describing zeros of homogeneous polynomials in . * Splitting of prime ideals in Galois extensions, Tamely ramified extensions of both fields are in bijection to one another. * Adjoining arbitrary -power roots of (in ), respectively of (in ), yields (infinite) extensions of these fields known as perfectoid fields. Strikingly, the Galois groups of these two fields are isomorphic, which is the first glimpse of a remarkable parallel between these two fields: \operatorname \left(\mathbf Q_p \left(p^ \right) \right) \cong \operatorname \left(\mathbf F_p((t))\left(t^\right)\right).


Differential fields

Differential fields are fields equipped with a derivation (abstract algebra), derivation, i.e., allow to take derivatives of elements in the field. For example, the field R(''X''), together with the standard derivative of polynomials forms a differential field. These fields are central to differential Galois theory, a variant of Galois theory dealing with linear differential equations.


Galois theory

Galois theory studies algebraic extensions of a field by studying the Symmetry group#Symmetry groups in general, symmetry in the arithmetic operations of addition and multiplication. An important notion in this area is that of finite extension, finite Galois extensions , which are, by definition, those that are separable extension, separable and normal extension, normal. The primitive element theorem shows that finite separable extensions are necessarily simple extension, simple, i.e., of the form :, where is an irreducible polynomial (as above). For such an extension, being normal and separable means that all zeros of are contained in and that has only simple zeros. The latter condition is always satisfied if has characteristic 0. For a finite Galois extension, the
Galois group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
is the group of field automorphisms of that are trivial on (i.e., the bijections that preserve addition and multiplication and that send elements of to themselves). The importance of this group stems from the fundamental theorem of Galois theory, which constructs an explicit one-to-one correspondence between the set of subgroups of and the set of intermediate extensions of the extension . By means of this correspondence, group-theoretic properties translate into facts about fields. For example, if the Galois group of a Galois extension as above is not solvable group, solvable (cannot be built from
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s), then the zeros of ''cannot'' be expressed in terms of addition, multiplication, and radicals, i.e., expressions involving \sqrt[n]. For example, the symmetric groups is not solvable for . Consequently, as can be shown, the zeros of the following polynomials are not expressible by sums, products, and radicals. For the latter polynomial, this fact is known as the Abel–Ruffini theorem: : (and ), : (where is regarded as a polynomial in , for some indeterminates , is any field, and ). The tensor product of fields is not usually a field. For example, a finite extension of degree is a Galois extension if and only if there is an isomorphism of -algebras :. This fact is the beginning of Grothendieck's Galois theory, a far-reaching extension of Galois theory applicable to algebro-geometric objects.


Invariants of fields

Basic invariants of a field include the characteristic and the transcendence degree of over its prime field. The latter is defined as the maximal number of elements in that are algebraically independent over the prime field. Two algebraically closed fields and are isomorphic precisely if these two data agree. This implies that any two uncountable algebraically closed fields of the same cardinality and the same characteristic are isomorphic. For example, and are isomorphic (but ''not'' isomorphic as topological fields).


Model theory of fields

In model theory, a branch of mathematical logic, two fields and are called elementarily equivalent if every mathematical statement that is true for is also true for and conversely. The mathematical statements in question are required to be first-order logic, first-order sentences (involving 0, 1, the addition and multiplication). A typical example, for , ''n'' an integer, is : = "any polynomial of degree in has a zero in " The set of such formulas for all expresses that is algebraically closed. The Lefschetz principle states that is elementarily equivalent to any algebraically closed field of characteristic zero. Moreover, any fixed statement holds in if and only if it holds in any algebraically closed field of sufficiently high characteristic. If is an ultrafilter on a set , and is a field for every in , the ultraproduct of the with respect to is a field. It is denoted by :, since it behaves in several ways as a limit of the fields : Łoś's theorem states that any first order statement that holds for all but finitely many , also holds for the ultraproduct. Applied to the above sentence , this shows that there is an isomorphismBoth and are algebraically closed by Łoś's theorem. For the same reason, they both have characteristic zero. Finally, they are both uncountable, so that they are isomorphic. :\operatorname_ \overline \mathbf F_p \cong \mathbf C. The Ax–Kochen theorem mentioned above also follows from this and an isomorphism of the ultraproducts (in both cases over all primes ) :. In addition, model theory also studies the logical properties of various other types of fields, such as real closed fields or exponential fields (which are equipped with an exponential function ).


The absolute Galois group

For fields that are not algebraically closed (or not separably closed), the absolute Galois group is fundamentally important: extending the case of finite Galois extensions outlined above, this group governs ''all'' finite separable extensions of . By elementary means, the group can be shown to be the Prüfer group, the profinite completion of . This statement subsumes the fact that the only algebraic extensions of are the fields for , and that the Galois groups of these finite extensions are given by :. A description in terms of generators and relations is also known for the Galois groups of -adic number fields (finite extensions of ). Galois representation, Representations of Galois groups and of related groups such as the Weil group are fundamental in many branches of arithmetic, such as the Langlands program. The cohomological study of such representations is done using Galois cohomology. For example, the Brauer group, which is classically defined as the group of central simple algebra, central simple -algebras, can be reinterpreted as a Galois cohomology group, namely :.


K-theory

Milnor K-theory is defined as :K_n^M(F) = F^\times \otimes \cdots \otimes F^\times / \left\langle x \otimes (1-x) \mid x \in F \setminus \ \right\rangle. The norm residue isomorphism theorem, proved around 2000 by Vladimir Voevodsky, relates this to Galois cohomology by means of an isomorphism :K_n^M(F) / p = H^n(F, \mu_l^). Algebraic K-theory is related to the group of invertible matrix, invertible matrices with coefficients the given field. For example, the process of taking the determinant (mathematics), determinant of an invertible matrix leads to an isomorphism K1(''F'') = ''F''×. Matsumoto's theorem (K-theory), Matsumoto's theorem shows that K2(''F'') agrees with K2M(''F''). In higher degrees, K-theory diverges from Milnor K-theory and remains hard to compute in general.


Applications


Linear algebra and commutative algebra

If , then the equation : has a unique solution in a field , namely x=a^b. This immediate consequence of the definition of a field is fundamental in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
. For example, it is an essential ingredient of Gaussian elimination and of the proof that any
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
has a basis (linear algebra), basis. The theory of module (mathematics), modules (the analogue of vector spaces over ring (mathematics), rings instead of fields) is much more complicated, because the above equation may have several or no solutions. In particular linear equation over a ring, systems of linear equations over a ring are much more difficult to solve than in the case of fields, even in the specially simple case of the ring \Z of the integers.


Finite fields: cryptography and coding theory

A widely applied cryptographic routine uses the fact that discrete exponentiation, i.e., computing : ( factors, for an integer ) in a (large) finite field can be performed much more efficiently than the discrete logarithm, which is the inverse operation, i.e., determining the solution to an equation :. In elliptic curve cryptography, the multiplication in a finite field is replaced by the operation of adding points on an elliptic curve, i.e., the solutions of an equation of the form :. Finite fields are also used in coding theory and combinatorics.


Geometry: field of functions

function (mathematics), Functions on a suitable topological space into a field can be added and multiplied pointwise, e.g., the product of two functions is defined by the product of their values within the domain: :. This makes these functions a -associative algebra, commutative algebra. For having a ''field'' of functions, one must consider algebras of functions that are integral domains. In this case the ratios of two functions, i.e., expressions of the form :\frac, form a field, called field of functions. This occurs in two main cases. When is a complex manifold . In this case, one considers the algebra of holomorphic functions, i.e., complex differentiable functions. Their ratios form the field of meromorphic functions on . The function field of an algebraic variety (a geometric object defined as the common zeros of polynomial equations) consists of ratios of regular functions, i.e., ratios of polynomial functions on the variety. The function field of the -dimensional affine space, space over a field is , i.e., the field consisting of ratios of polynomials in indeterminates. The function field of is the same as the one of any Zariski topology, open dense subvariety. In other words, the function field is insensitive to replacing by a (slightly) smaller subvariety. The function field is invariant under isomorphism and birational equivalence of varieties. It is therefore an important tool for the study of abstract algebraic variety, abstract algebraic varieties and for the classification of algebraic varieties. For example, the dimension of an algebraic variety, dimension, which equals the transcendence degree of , is invariant under birational equivalence. For algebraic curve, curves (i.e., the dimension is one), the function field is very close to : if is smooth variety, smooth and proper map, proper (the analogue of being compact topological space, compact), can be reconstructed, up to isomorphism, from its field of functions.More precisely, there is an equivalence of categories between smooth proper algebraic curves over an algebraically closed field and finite field extensions of . In higher dimension the function field remembers less, but still decisive information about . The study of function fields and their geometric meaning in higher dimensions is referred to as birational geometry. The minimal model program attempts to identify the simplest (in a certain precise sense) algebraic varieties with a prescribed function field.


Number theory: global fields

Global fields are in the limelight in
algebraic number theory Algebraic number theory is a branch of number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is th ...
and arithmetic geometry. They are, by definition, number fields (finite extensions of ) or function fields over (finite extensions of ). As for local fields, these two types of fields share several similar features, even though they are of characteristic 0 and positive characteristic, respectively. This function field analogy can help to shape mathematical expectations, often first by understanding questions about function fields, and later treating the number field case. The latter is often more difficult. For example, the Riemann hypothesis concerning the zeros of the Riemann zeta function (open as of 2017) can be regarded as being parallel to the Weil conjectures (proven in 1974 by Pierre Deligne). Cyclotomic fields are among the most intensely studied number fields. They are of the form , where is a primitive -th
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, i.e., a complex number satisfying and for all . For being a regular prime, Ernst Kummer, Kummer used cyclotomic fields to prove Fermat's Last Theorem, which asserts the non-existence of rational nonzero solutions to the equation :. Local fields are completions of global fields. Ostrowski's theorem asserts that the only completions of , a global field, are the local fields and . Studying arithmetic questions in global fields may sometimes be done by looking at the corresponding questions locally. This technique is called the local-global principle. For example, the Hasse–Minkowski theorem reduces the problem of finding rational solutions of quadratic equations to solving these equations in and , whose solutions can easily be described. Unlike for local fields, the Galois groups of global fields are not known. Inverse Galois theory studies the (unsolved) problem whether any finite group is the Galois group for some number field . Class field theory describes the abelian extensions, i.e., ones with abelian Galois group, or equivalently the abelianized Galois groups of global fields. A classical statement, the Kronecker–Weber theorem, describes the maximal abelian extension of : it is the field : obtained by adjoining all primitive -th roots of unity. Kronecker Jugendtraum, Kronecker's Jugendtraum asks for a similarly explicit description of of general number fields . For imaginary quadratic fields, F=\mathbf Q(\sqrt), , the theory of complex multiplication describes using elliptic curves. For general number fields, no such explicit description is known.


Related notions

In addition to the additional structure that fields may enjoy, fields admit various other related notions. Since in any field 0 ≠ 1, any field has at least two elements. Nonetheless, there is a concept of field with one element, which is suggested to be a limit of the finite fields , as tends to 1. In addition to division rings, there are various other weaker algebraic structures related to fields such as quasifields, Near-field (mathematics), near-fields and semifields. There are also proper classes with field structure, which are sometimes called Fields, with a capital F. The surreal numbers form a Field containing the reals, and would be a field except for the fact that they are a proper class, not a set. The nimbers, a concept from game theory, form such a Field as well.


Division rings

Dropping one or several axioms in the definition of a field leads to other algebraic structures. As was mentioned above, commutative rings satisfy all axioms of fields, except for multiplicative inverses. Dropping instead the condition that multiplication is commutative leads to the concept of a ''division ring'' or ''skew field''.Historically, division rings were sometimes referred to as fields, while fields were called ''commutative fields''. The only division rings that are finite-dimensional -vector spaces are itself, (which is a field), the quaternions (in which multiplication is non-commutative), and the octonions (in which multiplication is neither commutative nor associative). This fact was proved using methods of algebraic topology in 1958 by Michel Kervaire, Raoul Bott, and John Milnor. The non-existence of an odd-dimensional division algebra is more classical. It can be deduced from the hairy ball theorem illustrated at the right.


Notes


References

* * * , especially Chapter 13 * * * * * * . See especially Book 3 () and Book 6 (). * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * {{DEFAULTSORT:Field (Mathematics) Field (mathematics), Algebraic structures Abstract algebra